A Sample Final Exam

Math 1300Y Students: Make sure to write ``1300Y'' in
the course field on the exam notebook. Solve 2 of the 3 problems in
part A and 4 of the 6 problems in part B. Each problem is worth 17
points, to a maximal total grade of 102. If you solve more than the
required 2 in 3 and 4 in 6, indicate very clearly which problems you
want graded; otherwise random ones will be left out at grading and they
may be your best ones! You have 3 hours. No outside material other than
stationary is allowed.

Math 427S Students: Make sure to write ``427S'' in the
course field on the exam notebook. Solve 5 of the 6 problems in part B,
do not solve anything in part A. Each problem is worth 20 points. If
you solve more than the required 5 in 6, indicate very clearly which
problems you want graded; otherwise random ones will be left out at
grading and they may be your best ones! You have 3 hours. No outside
material other than stationary is allowed.

Good Luck!

Part A

Problem 1. Let be a topological space.

Define the ``product topology'' on .

Prove that if is compact then so is .

Prove that the ``folding of along the diagonal'',
is also compact.

Problem 2. Let be a compact
metric space and let
be an open cover of
. Show that there exists
such that for every
there exists
such that the -ball centred at
is contained in . ( is called a Lebesgue
number for the covering.)

Problem 3.

Compute
.

A topological space is obtained from a topological space
by gluing to an -dimensional cell using a continuous
gluing map
, where . Prove that
obvious map
is an isomorphism.

Compute
for all .

Part B

Problem 4. Let be a covering of a
connected locally connected and semi-locally simply connected base
with basepoint . Prove that if
is normal in
then the group of automorphisms of acts transitively on
.

Problem 5. A topological space
is obtained from a topological space by gluing to an
-dimensional cell using a continuous gluing map
, where . Show that

for
.

There is an exact sequence

Problem 6. Let denote the (standard)
2-dimensional torus.

State the homology and cohomology of including the
ring structure. (Just state the results; no justification is required.)

Show that every map from the sphere to induces the zero
map on cohomology. (Hint: cohomology flows against the direction of ).

Problem 7. For , what is the
degree of the antipodal map on ? Give an example of a continuous
map
of degree 2 (your exmple should work for every
). Explain your answers.

Problem 8.

State the ``Salad Bowl Theorem''.

State the ``Borsuk-Ulam Theorem''.

Prove that the latter implies the former.

Problem 9. Suppose

is a commutative diagram of Abelian groups in which the rows are exact and
, , and are isomorphisms. Prove that
is also an isomorphism.

Good Luck!

Warning: The real exam will be similar to this sample, to
my taste. Your taste may be significantly different.